A point P moves on the parabola x2 – 2x + 4y – 11 = 0. The equation of locus of mid-point of line segment PS is (where S is the focus of the parabola)

Question

A point P moves on the parabola x² – 2x + 4y – 11 = 0. The equation of the locus of mid-point of line segment PS is (where S is the focus of the parabola)

Answer 1

x² – 2x + 4y – 11 = 0
⇒ x² – 2x – 11 = -4y 
⇒ x² – 2x + 1 - 12 = -4y 
⇒ (x - 1)² = 4(-1)(y - 3) 
From here we can identify the focus.
As in (x - h)² = 4p (y - k) the focus is given by (h, k + p)
The directrix is y = k - p
Here focus is (1, 4) and directrix is y = 4

Now it's easy to solve as the locus formed by that point will also be a parabola with same focus and distance will be half.